Variational principle for contact Hamiltonian systems and its applications

In [8], the authors provided an implicit variational principle for the contact Hamilton's equations {x˙=[Formula presented](x,u,p),p˙=−[Formula presented](x,u,p)−[Formula presented](x,u,p)p,(x,p,u)∈T^⁎M×R,u˙=[Formula presented](x,u,p)⋅p−H(x,u,p), where M is a closed, connected and smooth manifold and H=H(x,u,p) is strictly convex, superlinear in p and Lipschitz in u. In the present paper, we focus on two applications of the variational principle: 1. We provide a representation formula for the solution semigroup of the evolutionary equation w_t(x,t)+H(x,w(x,t),w_x(x,t))=0; 2. We study the ergodic problem of the stationary equation via the solution semigroup. More precisely, we find pairs (u,c) with u∈C(M,R) and c∈R which, in the viscosity sense, satisfy the stationary partial differential equation H(x,u(x),u_x(x))=c.